Subgroups — #2276×C₂₁₁ #41009896

Order 27008.

628 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₂₁₁	C₂	1	3	yes	maximal
2	C₄×C₄⋊C₄×C₂₁₁	C₂	1	9	yes	maximal
3	#376×C₂₁₁	C₂	1	3	yes	maximal
4	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
5	C₄×Q₈×C₂₁₁	C₄	1	24	yes	complemented
6	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	9	yes	—
7	C₂×C₄²×C₂₁₁	C₂²	1	6	yes	—
8	C₂×C₄²×C₂₁₁	C₂²	1	9	yes	—
9	#95×C₂₁₁	C₂²	1	1	yes	characteristic
10	#95×C₂₁₁	C₂²	1	9	yes	—
11	C₂×Q₈×C₂₁₁	C₂×C₄	1	12	yes	—
12	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
13	C₂²×C₄×C₂₁₁	C₂³	1	9	yes	—
14	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
15	C₄⋊C₄×C₂₁₁	C₂×C₄	1	36	yes	—
16	C₄²×C₂₁₁	—	4	6	no	—
17	C₄²×C₂₁₁	C₂×C₄	1	36	yes	—
18	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
19	Q₈×C₂₁₁	C₄²	1	16	yes	complemented
20	C₂×C₄×C₂₁₁	—	2	9	no	—
21	C₂×C₄×C₂₁₁	—	2	6	no	—
22	C₂×C₄×C₂₁₁	—	2	9	no	—
23	C₂×C₄×C₂₁₁	C₂²×C₄	1	6	yes	—
24	C₂×C₄×C₂₁₁	C₂²×C₄	1	18	yes	—
25	C₂×C₄×C₂₁₁	C₂²×C₄	1	18	yes	—
26	C₂²×C₂₁₁	C₂³×C₄	1	3	yes	—
27	C₂²×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
28	C₂²×C₂₁₁	2₋¹⁺⁴	1	3	yes	—
29	C₄×C₂₁₁	—	2	6	no	—
30	C₄×C₂₁₁	—	2	18	no	—
31	C₄×C₂₁₁	C₂×C₄²	1	12	yes	—
32	C₂×C₂₁₁	C₂²×C₄²	1	1	yes	characteristic
33	C₂×C₂₁₁	#519	1	3	yes	—
34	C₂×C₂₁₁	#520	1	3	yes	—
35	C₂₁₁	#2276	1	1	yes	characteristic, direct product factor, Sylow
36	#2276	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₄×Q₈	C₂×C₂₁₁	1	3	yes	—
38	C₄×C₄⋊C₄	C₂×C₂₁₁	1	9	yes	—
39	#376	C₂×C₂₁₁	1	3	yes	—
40	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
41	C₄×Q₈	C₄×C₂₁₁	1	24	yes	complemented
42	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	9	yes	—
43	C₂×C₄²	C₂²×C₂₁₁	1	6	yes	—
44	C₂×C₄²	C₂²×C₂₁₁	1	9	yes	—
45	#95	C₂²×C₂₁₁	1	1	yes	characteristic
46	#95	C₂²×C₂₁₁	1	9	yes	—
47	C₂×Q₈	C₂×C₄×C₂₁₁	1	12	yes	—
48	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
49	C₂²×C₄	C₂³×C₂₁₁	1	9	yes	—
50	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
51	C₄⋊C₄	C₂×C₄×C₂₁₁	1	36	yes	—
52	C₄²	—	4	6	no	—
53	C₄²	C₂×C₄×C₂₁₁	1	36	yes	—
54	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
55	Q₈	C₄²×C₂₁₁	1	16	yes	complemented
56	C₂×C₄	—	2	9	no	—
57	C₂×C₄	—	2	6	no	—
58	C₂×C₄	—	2	9	no	—
59	C₂×C₄	C₂²×C₄×C₂₁₁	1	6	yes	—
60	C₂×C₄	C₂²×C₄×C₂₁₁	1	18	yes	—
61	C₂×C₄	C₂²×C₄×C₂₁₁	1	18	yes	—
62	C₂²	C₂³×C₄×C₂₁₁	1	3	yes	—
63	C₂²	H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic
64	C₂²	2₋¹⁺⁴×C₂₁₁	1	3	yes	—
65	C₄	—	2	6	no	—
66	C₄	—	2	18	no	—
67	C₄	C₂×C₄²×C₂₁₁	1	12	yes	—
68	C₂	C₂²×C₄²×C₂₁₁	1	1	yes	characteristic
69	C₂	#519×C₂₁₁	1	3	yes	—
70	C₂	#520×C₂₁₁	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂⁴
commutator	C₂	C₂²×C₄²×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. C₄×Q₈×C₂₁₁	C₄	semidirect
19. Q₈×C₂₁₁	C₄²	semidirect
35. C₂₁₁	#2276	direct (N × H)
36. #2276	C₂₁₁	direct (N × H)
41. C₄×Q₈	C₄×C₂₁₁	semidirect
55. Q₈	C₄²×C₂₁₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×Q₈×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
#376×C₂₁₁	2
#2276	211

Subgroups — #2276×C211 #41009896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2276×C₂₁₁ #41009896